ROTACIONAL Y DIVERGENCIA DE UN CAMPO VECTORIAL

Oh i sorry i think that repeat the same concept that my classmate Bryan said.

Before giving my opinion about the topics of the forum i wanted to clear up the concept of vectorial field

Campos vectoriales

Vectorial fields vectorial Fields. A vectorial field in Rn is a function F: W! Rn where W is Rn's subset that usually will be opened. Therefore, a vectorial field has n coordinates, which are fields you will climb; concretly, for x = (x1, x2..., xn) 2 W, the vector F (x) 2 Rn will have to have the form
F(x,y) =P(x,y),Q(x,y)= P(x,y) i + Q(x,y) j

Rotacion de un campo vectorial
Rotation of a vectorial field is defined as a vectorial field whose magnitud is the maximum traffic for the unit of area in every point of the space whose direction and sense are those of the normal vector to the environment that maximizes the traffic in every point of the space

Rotation of a vectorial field
is defined in a point P as the clear one of the field by unit of volume evaluated in the above mentioned point, it appears to study the behavior of the fields and his sources

integrales de linea
lineal integrals is the sum of differential elements taken through contour which is called: way of integration or contour of integration An lineal integral is conformed by the followings Elements: the symbol of the integral of line, the field that joins can be (scalar or vectorially) and the operator that multiplies the field by the differential one of length.

Hello this example is very good.
Example:
Be: F(x,y,z)= X2 yz i + 3xyz3j + (x2-z2) k
Calculate dif. F and Rot F solution
dif. F= grad *F= 2xyz + 3xz3 - 2z
= -(9xyz2) i - (2x-x2y) + (3yz3 - x2z) k

Examples of this important issue are very essential to know because we understand development thematic field forces in Conserve as f can be expressed as a function gradientede potential g (x, y, z,)

The gradient operator = (/ x) + (/ y) + (/ z) can be combined with a vector function of two additional ways.


Rotational Properties.
1. If the scalar field f (x, y, z) has continuous partial derivatives of second order
then the curl (∇ f) = 0

Example

Find the potential function for the conservative vector field in the last example

G = (1 + 2xy) i + (x 2 - 2) j

Solution Solution

We have We have

M = 1 + 2xy = f x


Integrating both sides with respect to x we get


x + x 2 y + c(y) = f(x,y) x + x 2 y + c (y) = f (x, y)

Notice that the constant of integration may involve y terms since we are treating y as a constant. Now differentiate with respect to y to get

x 2 + c'(y) = f y = N = x 2 - 2 x 2 + c '(y) = f y = N = x 2 to 2

Thus Thus

c'(y) = -2 c '(y) = -2

Integrating with respect to y, we get

c(y) = -2y c (y) =-2y

We do not need a constant of integration here since we just want "a" potential function not the general potential function. Putting it all together, we get the potential function

f(x,y) = x + x 2 y - 2y f (x, y) = x + x 2 y - 2y

2. If F (x, y, z) is a conservative vector field then curl (F) = 0

There is defined the difference of a vectorial field in a point as the limit where the limit takes on volumes  increasingly small that tend to the point. The difference of a vectorial field is a quantity to climb. This quantity is independent from the succession of volumes that take so as that they converge on the same point of a uniform way.

We have seen that if a field of forces is conserving, at the time can express as the gradient of a potential function g(x,y,z).

= (g/ x) + (g/y) + (g/ z)
= Grad g(x, y, z) = g (x, y, z)
The active gradient, = ( / x) + ( / y) + ( / z) It is possible to combine with a vectorial function of two additional ways.

The ROTACIONAL of a vectorial field
= P + Q + R
It is the vector

rot = X

Good summary All this FOR MY PARTNER I GET TO THE CONCLUSION THAN.

Also, you can narrow the domain if irrotoacional, but not simply connected.

Assuming F (x, y) = (x / [(x ^ 2 + y ^ 2)] ^ 1 / 2, y / [(x ^ 2 + y ^ 2) ^ 1 / 2])

f2'x - f1'y =-x / [(x ^ 2 + y ^ 2) ^ 3 / 2] + x / [(x ^ 2 + y ^ 2) ^ 3 / 2] = 0

However, the domain of F is F: R2 - (0,0) -> R2

If the field is conservative by Green's Theorem, the rotor case in R2, you know that the integral over any surface surrounding the point is constant and not surround any point is 0.

To say that is conservative (and there is a potential F), enough to exclude the formation of a curve around 0, ie Dom (F): y> 0.


GRACIAS

definition of curl.

Let F be a vector field given by F : D ⊂ R3 →R3 / F(X,Y,Z) = F1((X,Y,Z);F2(X,Y,Z);F3(X,Y,Z)).
have F1;F2;F3.continuous partial derivatives in some region R. The curl of the field F is
given by
rot = ∇ × f= (∂F3/∂Y - ∂F2/∂Z)i+(∂F1/∂z - ∂F3/∂x)j+(∂F2/∂x - ∂F1/∂y)k

is important that their properties for better understanding.

Rotational Properties.

1. If the scalar field f (x, y, z) has continuous partial derivatives of second order
then the curl (∇ f) = 0


2. If F (x, y, z) is a conservative vector field then curl (F) = 0


3. If the vector field F (x, y, z) is a function defined on all 3 of which
components have continuous partial derivatives and curl (F) = 0

then F is a
conservative vector field.

a vector field that is obtained by calculating the curl of a field f at each point.

is known as vector fonts (where the scalar sources which are obtained by divergence).

A field whose curl is zero at all points in space is called irrotational or said that no vector fonts. And if it is defined on a simply connected domain then this field can be expressed as the gradient of a scalar function

Here, ΔS is the surface area resting on the curve C, which reduces to a point. The result of this rotational limit is not complete (which is a vector), but only its component along the direction normal to ΔS and oriented according to the right hand rule. To get the full rotational three limits should be calculated considering three corners located in perpendicular planes.

Although the curl of a field about a point is not zero does not imply that the field lines rotate around that point and encieren. For example, the velocity field of a fluid flowing through a pipe (known as Poiseuille profile) has a non-zero curl everywhere except the central axis, although the current flows in a straight line: This example unfortunately not shows its demonstration.

The curl of a vector field has its main physical interpretation when the
vector function F (x, y, z) represents the fluid flow, the curl in this case
interpreted as presenting the fluid flow around a point (Xo,Yo,Zo)
If the vector field F represents the fluid flow and curl (F) = 0

then we say
that the fluid is irrotational.


EXAMPLE: Let the vector field F (x, y, z) = (0, cos (xz),-sin (x)) determines
its curl.

Solution. In applying the definition of the curl yields the following vector that
represents:

rot (F)= (∂/∂y(-sen(xy))-∂/∂z(cos(xy)))i+(∂/∂z(0)-∂/∂x(-sen(xy)))j+(∂/∂x(cos(xz))-∂/∂/y(0)))k
=(-xcos(xy)+xsen(xz))i+(ycos(xy))j+(-zsen(xz))k
=x(sen(xz)-cos(xy))i+ycos(xy)j-zsen(xz)k

Example
Show that if F (x, y) = (2x + y3) i + (3xy2 + 4) j, then ∫ c F. dr is independent of the path and evaluate ∫ F. dr

Solution.
The vector function F have continuous first partial derivatives for all (x, y) and therefore can apply Theorem (18.16). Taking M = 2x + y3 and N = +4 we see 3xy2

∂ ∂ M_ = 3y2 = N_

∂ y ∂ x

Therefore, the line integral is independent of the path. By Theorem (18.13), there exists a function (potential) f such that

fx (x, y) = 2x + y3 and f (x, y) = 3xy2 + 4.

If we integrate (partially) fx (x, y) with respect to x,

fx (x, y) = x2 + xy3 + k (y)

where k is a function only of y. (You have to use k (and) instead of a constant in the partial integration to obtain the general expression of f (x, y) such that fx (x, y) = 2x + y3.)

Differentiating f (x, y) with respect to y, and comparing with the expression f (x, y) = +4 obtain 3xy2

f (x, y) = 3xy2 + k '(y) = 3xy2 + 4.

Therefore, k '(y) = 4 or k (y) = 4y + c for some constant c. Then

f (x, y) = x2 + xy3 + 4y + c

defines a function of the desired type. Applying Theorem (18.14),

∫ (2x + y3) dx + (3xy2 + 4) dy = x2 + xy3 + 4y]

= (4 + 54 + 12) - 4 = 66

Green's theorem

∫ f '(x) dx = f (b) - f (a)

He says that the integral of a function over a set S = [a, b] is equal to a related function (the antiderivative) evaluated in some way on the boundary of S, in this case consists of just two points, and b.

Theorem A
Let C be a simple closed curve, piecewise smooth, which forms the border of a region S xy plane. If M (x, y) and N (x, y) are continuous and have continuous derivatives on its border and C, then

N_ ∂ - ∂ M_ dA = M dx + N dy

∂ x ∂ y

s

Proof. Prove the theorem for the case where S is simple as both x-and-simple and then discuss extensions to the general case. Since S is y-simple, t, ie

S = ((x, y): g (x) ≤ y ≤ f (x), a ≤ x ≤ b)

Example for understanding the applications of the theorems
Show that if a region S of the plane has the border into C, this being a simple piecewise smooth curve and closed, then the area of S is given by



A (S) = ½ x dy - y dx

Solution.
Let M (x, y) =-y / 2 and N (x, y) = x / 2 and apply Green's theorem.



- Y dx + x dy = ∫ ∫ dA 1 + 1 = A (S)

2 2 2 2

other examples of vector field when the distribution of TRENGTHENING semodela a estrutura, the distribution of forces or gravitational electromanectica nature in space is done using vector fields.
Velocity functions associated with the trajectories of particles or differential volume of a substance flow conditions either laminar or turbulent.

The gradient of a scalar field, is another example of vector field, because the magnitude and direction of the gradient of a scalar field is a function of the coordinates

Let F be a vector field defined by
F: R3 → R3 / F (x, y, z) = (F1( x, y, z), F2 (x, y, z,) F3 (x, y, z,) where F1,F2,F3

have continuous partial derivatives in some region R. The divergence of F is denoted
by div F, or ∇. F, and is given by.

div (F)= ∇.F = ∂F1/∂X + ∂F2/∂Y + ∂F3/∂Z.


Divergence Properties.

1. If F(x, y, z) = (F1( x, y, z,) F2 (x, y, z,) F3 (x, y, z)) is a vector field on R3, and
F1, F2 and F3 have continuous partial derivatives of second order then
div (curl (F)) = 0.
2. If f (x, y, z) is a scalar field, the divergence of its gradient vector field
div (∇.f), is given by.

div (∇.f)= ∇.∇.f= ∂2F/∂X2 + ∂2F/∂Y2 + ∂2F/∂Z2.


An expression that is often shortened by ∇2 f, where the operator ∇2 f, is called
as the Laplace operator. This operator can also be used to a field
F(x, y, z) = (F1( x, y, z,) F2 (x, y, z,) F3 (x, y, z)) applying it to each of its
component functions, ie.

∇2 f= (∇2 f1,∇2 f2,∇2 f3)

In fluid mechanics, if the fluid velocity field is given by the
vector field F, is the div (F) = 0 is said that the fluid is incompressible.

Annex an example

EXAMPLE Let the vector field F (x, y, z) = (ex sen (y),ex cos (y), z) determines
their divergence.

Solution.

div (∇.f)= ∂/∂X(ex sen(y))+ ∂/∂y (ex cos(y)) + ∂/∂z (z)
= ex sen(y)- ex sen(y)+ 1
= 1

In the subject we are talking about developing another type of vector functions to which we already know, these 2 new vector functions are functions that assign a vector to a point in the plane or a point in space. Such functions are called vector fields and are useful to represent different force fields and velocity fields. Can be expressed through the functions F defined by:
F (x, y) = Mi + Nj; in the plane.
Where M and N are functions of two variables X and Y, defined in a plane region R.
F (x, y, z) = Mi + Nj + Pk; in space.
Where: M, N and P are functions of three variables X, Y and Z, Q defined in a region of space.
Therefore, we see that the gradient is an example of vector field, since it can be written in the form:
▼ f (x, y) = i + (x, y) j Ruth Jimenez.
or ▼ f (x, y, z) = i + (x, y, z) j + (x, y, z) k
(Recall that f ▼ is only differentiation with respect to each variable in a function)

first define gradient.

The gradient usually indicates a direction in space which is seen as a variation of a particular property or physical quantity.

In other contexts gradient is used informally to indicate the existence of progressive or gradual variation in a specific area, not necessarily related to the physical layout of a particular size or ownership.

Now the gradient vector field.

Gradient of a vector field in Euclidean space, the concept of gradient also can be extended to a vector field, being the gradient of a tensor which gives the differential of the field to perform a shift


DF = F (r + dr) - f (r) = (gradient.F).dr f and r are vectors

This tensor can be represented by a matrix, which in Cartesian coordinates is made up of three partial derivatives of the three components of the vector field

taking into account the definition of partner and Green's theorem relating the divergence theorem states that the theorem is equivalent to the double integral evaluated at the boundaries of D gradient of the function F dA is equal to the integral C evaluated by the unit vector function F n ds where n is the unit vector or unit vector also called the module of this vector raised to the root of dx 2 more dy elevaod the resulting cuaddrado ds. taking these data F = <P,Q> Convert the equation: the integral evaluated at C of the function F for n ds is equal to the integral of less QDX PDY
that through the green theorem is
integal evaluated the C-PDY QDX more equal to the double integral evaluated at D of the derivative of P with respect to x over the derivative of Q with respect to y of dA equals the double integral evaluated in the limits of the gradient D function F from dA

Example of divergence
Show that any vector field defined by
F (x, y, z) = (f (y, z), g (x, z), h (x, y)) is incompressible.
Solution.

Div (F) = ∂ / ∂ x (f (z)) + ∂ / ∂ y (f (xz)) + ∂ / ∂ z (f (x)


= 0

In fluid mechanics, if the fluid velocity field is given by the
vector field F, is the div (F) = 0 is said that the fluid is incompressible

example of a vector field rotational

Determine if the vector field defined by
F (x, y, z) = (2xy, x2 + 2yz, y2) is a conservative field.
Solution. For rotational properties of a vector field is conservative if
Rot (F) = 0

To demonstrate applications of the curl definition to calculate it.
Where it is proved that F (x, y, z) = (2xy, x2 + 2yz, y2) is a field
Conservative.
Rot (F) = ∂ / ∂ x (y2) ∂ / ∂ z (x2 +2 z)) i + ∂ / ∂ z (2xy) - ∂ / ∂ z (y2)) j + ∂ / ∂ x (x2 + 2yz) - ∂ / ∂ y (2xy)) k
= (2y-2y) i + (0 +0) j + (2x-2x) k
= 0i +0 j +0 k

Where it is proved that F (x, y, z) = (2xy, x2 + 2yz, y2) is a field
conservative.